Thermodynamics of gold nanoparticle growth

ermodynamics of gold nanoparticle growth:A first-principles investigation

esis for the degree of Erasmus MundusMaster of Science in Nanoscience and Nanotechnology

HIMANSHU SHEKHARDepartment of Applied Physics,C U T,G, S 2013

esis for the degree of Erasmus MundusMaster of Science in Nanoscience and Nanotechnology


Supervisor:Paul Erhart, Chalmers University of Technology, Sweden

Co-supervisor:Steven De Feyter, K U Leuven, Belgium

External examiner:Kasper Moth-Poulsen, Chalmers University of Technology, Sweden

Department of Applied Physics,CHALMERS UNIVERSITY OF TECHNOLOGY

Göteborg, Sweden 2013

is master thesis is conducted in the framework of Erasmus Mundus MasterProgramme in Nanoscience and Nanotechnology


Himanshu Shekhar

©Himanshu Shekhar, 2013

Department of Applied Physics,CHALMERS UNIVERSITY OF TECHNOLOGYSE 412-96, Göteborg, Sweden 2013

Printed by Chalmers reproserviceGöteborg, Sweden 2013

Abstract

Good control over size and shapes of nanoparticles is key for their usageas basic building blocks in various applications. Nanocrystals are of immenseinterest because of their electronic, optical, and magnetic properties, which aredependent on size and morphology of the nanocrystals. Hence by controllingthese parameters it is possible to tune these properties as well. In spite of sig-nificant progress in the area of crystallography and materials science the un-derlying growth mechanism of anisotropic nanocrystals has not been resolvedto a satisfactory level. Proper understanding of growth and control mechanismis key for further improvement, which will open the door for applications innew areas.

Gold nanorods are examples for anisotropic nanocrystals which is a topicof research for many groups. Boom-up , e.g., templating, electrochemicaland top-down, e.g., photolithography are two methods for the synthesis ofmetal nanoparticles. One of the most widely used methods for the synthesisof anisotropic gold nanorods is surfactant assisted seed mediated growth fromsolution. Seed mediated growth is a wet chemical method is a cheap and easyprocess. While it offers several advantages over electrochemical processes, itsuffers from low yield and the unpredictible morphology of the end product inmany cases.

e objective of this thesis is to resolve the thermodynamics of the growthof Au nanoparticles and rods using cetyltrimethylammoniumbromide as growthagent. To this end the surface energetics of a variety of different Au surfaceswith and without surfactant are obtained using electronic structure calcula-tions based on density functional theory. e results are discussed in the con-text of experimental observations. e growth of nanocrystals is not an equi-librium phenomena so its growthmechanism cannot be explained by pure ther-modynamics alone. e present study provides basic thermodynamic param-eters, which determine the driving forces and exact knowledge of which is aprerequisite for understanding the non-equilibrium growth process. erebypresent work lays the ground for future studies that should address, for ex-ample, entropic and kinetic factors in order to get a comprehensive picture ofnanocrystal and nanorod formation.

Anowledgements

e work presented in this thesis was carried out at the Applied Physics de-partment at Chalmers University, in the year 2013. It is now my responsibilityand pleasure to express my gratitude to the people involved in this work.

I greatly acknowledge my supervisor Asst. Prof. Paul Erhart for acceptingme for this master thesis project. Paul’s trust, patience, and knowledge madethis journey possible. I am grateful to Prof. Steven De Feyter for co-supervisingme. I am grateful to Asst. Prof. Kasper Moth-Poulsen for first introducing meto this project and later for accepting the role of referee in my thesis defense.

I recognize the help of all PhD students of Materials and Surface theorygroup at Chalmers university. I would also like to thank my program coordi-nators Prof. Guido Groenseneken and Prof. Göran Johansson for their roles.Finally, I would like to thank my EMM Nano classmates with whom I spent agreat time of my life during this master program.

Contents

1 Introduction 11.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 Scope and Outline . . . . . . . . . . . . . . . . . . . . . . . . . 3

2 Methodology 52.1 Density Functional eory . . . . . . . . . . . . . . . . . . . . 62.2 Exchange-Correlation Functional . . . . . . . . . . . . . . . . 72.3 Choice of Basis Set . . . . . . . . . . . . . . . . . . . . . . . . 82.4 Pseudopotentials . . . . . . . . . . . . . . . . . . . . . . . . . . 102.5 Brillouin Zone Integration . . . . . . . . . . . . . . . . . . . . 102.6 Slab Models of Surfaces . . . . . . . . . . . . . . . . . . . . . . 112.7 Computational Setup . . . . . . . . . . . . . . . . . . . . . . . 12

3 ermodynamics 133.1 ermodynamics versus Kinetics . . . . . . . . . . . . . . . . 133.2 ermodynamic Driving Forces . . . . . . . . . . . . . . . . . 153.3 Surface Free Energy . . . . . . . . . . . . . . . . . . . . . . . . 163.4 ermodynamics of Adsorption . . . . . . . . . . . . . . . . . 17

3.4.1 Adsorbates . . . . . . . . . . . . . . . . . . . . . . . . 173.4.2 Approximations . . . . . . . . . . . . . . . . . . . . . . 183.4.3 Adsorption Sites . . . . . . . . . . . . . . . . . . . . . 19

3.5 Coverage . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 203.6 Equilibrium Shape . . . . . . . . . . . . . . . . . . . . . . . . . 21

4 Results and Discussion 234.1 Adsorption . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

4.1.1 Adsorption Sites . . . . . . . . . . . . . . . . . . . . . 234.1.2 Br on Low Index Au Surfaces . . . . . . . . . . . . . . 234.1.3 BTA+ on Low Index Au Surfaces . . . . . . . . . . . . 254.1.4 BTAB on Low Index Surfaces . . . . . . . . . . . . . . 264.1.5 Br on High Index Surface . . . . . . . . . . . . . . . . 28

I

4.1.6 BTA+ on High Index Surface . . . . . . . . . . . . . . 284.1.7 BTAB on High Index Surface . . . . . . . . . . . . . . 29

4.2 Surface Energy . . . . . . . . . . . . . . . . . . . . . . . . . . . 314.2.1 Surface Energies of Clean Au Surfaces . . . . . . . . . 314.2.2 Surface Energies Change due to Adsorption . . . . . . 31

4.3 Equilibrium Crystal Shape . . . . . . . . . . . . . . . . . . . . 35

5 Discussion and Conclusion 415.1 Template Assisted Growth . . . . . . . . . . . . . . . . . . . . 41

5.1.1 Chemical Potential . . . . . . . . . . . . . . . . . . . . 415.1.2 Anisotropic Shapes and Symmetry Breaking . . . . . . 42

5.2 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44

Bibliography 47

II

Chapter 1

Introduction

1.1 Baground

Most of us must have seen ice, a form of crystal, in various sizes and shapes.Why do they form from liquid? Why do they grow in different shapes? Whydoes a flake of ice grow differently on a surface of water in a pond than theone in our freezer? e first question is not difficult to answer. Ice growsfrom liquid because it lowers its free energy by doing so. e second questionis, however, not so easy to answer. One can make an educated guess that it,probably temperature, the surface onwhich the ice grows, and the environmentwhich dictate the final shapes of ice flakes. e details, however, are as usualmore complicated.

(a) (b)

Figure 1.1: (a) Calcidiscus, a biomineralized crystalline product (from the Interna-tional Nanoplankton Association website). (b) Nano-Flower, a germenium-sulfidenanosheet structure (from the Materials Research Society).

An intriguing branch of materials science is concerned with resolving thegrowth mechanism of organic and inorganic nanocrystals. It is interesting to

1

Chapter 1. Introduction

know that humans are not the first species, which envisaged the potential ap-plication of nanocrystals. Many living organisms in nature use biomineraliza-tion processes to build shells such as these shown in Fig. 1.1(a) and many otherpaerned structures. ese organisms are able to control nucleation, crystalshape, and material structure. What is most striking is the degree to whichthese organisms are able to control the growth of crystallites and the crys-tallographic orientation of its functional material compared to syntheticallyproduced crystals.

Taking a cue from this beautiful work of nature there have been continuousefforts see Fig. 1.1(b), to imitate it in the laboratory for its potential applicationin fields of electronics, bioimaging, biosensing, and opticas. Gold nanorodsare of particular interest for their potential use in biosensing and bioimagingmainly because of the inertness of gold metal in biological environments [1].In spite of the enormous success in growing nanocrystallites and nanorods ofvarious shapes the details of the growth process are still incompletely under-stood. A good example is surfactant assisted seedmediated growth of nanorods[2, 3, 4, 5], which also is the subject of this thesis. Several research groups haveadopted the seed mediated approach in aqueous solution using cetyltrimethy-lammonium bromide (CTAB) as surfactant to produce gold nanorods.

Figure 1.2: Growth of gold nanorods using CTAB bilayer as template. Binding ofCTAB to Au (100) facet and growth along (111) [2].

ere is, however, still ambiguity when it comes to describing the growthmechanism. e role of surfactants is not clearly known. e general tendencyin the literature is to emphasize the role of surfactants as shape directing agent.First Nikkobakht and later Murphy and co-workers proposed a zipping mech-anism [6, 2], Fig. 1.2, for nanorod like growth. In the zipping mechanism thesurfactants starts forming a bilayer in the solution at the critical micelle con-centration. ese bilayers act like a template on which incoming monomersaach and form a rod like shape. Others [7] argued that a bilayer structureformation is not necessary for the growth of nanorods. e surfactants arenormaly bigger in size compared to the monomer units of crystals thus stericeffects are also important. While surfactants thus definitely play an impor-

2

1.2. Scope and Outline

tant role in the growth of anisotropic nanocrystals, the exact role they play islargely unknown.

1.2 Scope and Outline

e objective of this thesis is to resolve the thermodynamics of growth of goldnanoparticles and predict their equilibrium shape under different conditions.All calculations in this work are based on the assumption that the nanocrystal-lite is in eqilibriumwith the surrounding medium. We are only concerned withthe surface and neglect any bulk effects such as strain. We have also neglectedcorner and edge effects.

e first step towards achieving this goal is to find a set of computationalparameters that ensure numerically converged calculations. Using the thusdetermined parameters the surface free energies of bare gold surfaces are com-puted to establish their relative stability. Motivated by the structural infor-mation on gold nanorods available in the literature [8, 9, 5], several surfacesof face centered cubic (FCC) gold crystal have been considered in this project.Surfactants change the surface free energy. Here we consider butyltrimethy-lammonium bromide (BTAB) as a surrogate for CTAB. Aer finding the equlib-rium positions the surface energy is calculated at different surface coverages.e thus obtained surface energies are then being used to predict equlibriumshapes using the Wulffconstruction method.

All computations are done within density functional theory (DFT) [10] us-ing the projector augmented wave method as implemented in vienna ab-initiosimulation package [11]. Chapter 2 will discuss basics of DFT with some com-ments on its accuracy in the present context. Chapter 3 is devoted to thermo-dynamics, its influence on gold nanorod growth and on the equilibrium shapeof nanocrystallites. Results are compiled in chapter 4 and finally this thesisconcludes with discussion and some final remarks.

3

Chapter 1. Introduction

4

Chapter 2

Methodology

To calculate the structure and electronic properties of molecules and solids re-quires solving the Schrödinger equation. For most systems it is impossible tosolve the quantum many-body problem analytically. A number of numericaltechniques have been formulated to solve such problems. ese techniques in-clude methods such as hartree-fock, quantum monte carlo [12], density func-tional theory [10]. e primary purpose of any ab-initio calculation is to char-acterize a material without any experimental or empirical input. Atomic posi-tions and atomic numbers are in principle the sole input parameters in thesemethods. With recent developments in theoretical techniques and the continu-ous increase in computational power, ab-initio techniques are not only limitedto the structural study of materials but also being used for example in the studyof defects in materials, adsorption on surfaces, and other surface related phe-nomena. Ab-initio calculations not just give information about the ideal stateof a system but can also help understand the impact of temperature, pressure,and chemical composition. ese studies have helped our understanding ofnew materials and lead to further insights and predictions, which would benearly impossible to achieve experimentally.

e following sections provide a brief overview of density functional theory(DFT) and its practical aspects, such as pseudopotentials, exchange-correlationfunctionals, and Brillouin-zone sampling. We also introduce the slab model ofthe gold surface, which will be our basic platform for all subsequent computa-tions. We then finish the chapter by describing the periodic superstructure ofadsorbates on gold surfaces.

5

Chapter 2. Methodology

2.1 Density Functional eory

e objective of DFT is to obtain ground state properties of any system in anexternal potential. DFT is based on set of approximations, which simplify theoriginal many body time independent Schrödinger equation,[

−ℏ2

2m

N∑i=1

∇2 +N∑i=1

V (ri) +N∑i=1

∑j<i

U(ri, rj)

]ψ = Eψ, (2.1)

where the first term on the le side represents the kinetic energy, the secondterm is the interaction between electrons and the external potential, and thethird term corresponds to the interaction between electrons.

e Hamiltonian depends on the spatial coordinates, ri, of each electron inthe system. It represents a formidable task to solve above equation numericallyeven for a small number of electrons. A breakthrough came in the mid 1960ieswhen Hohenberg and Kohn [13] proved that the energy of a system is a uniquefunctional of the electron density

E = F [n(r)], (2.2)

whereE is the energy and n(r) is the ground state electron density. e secondtheorem states that the electron density, which minimizes the ground stateenergy, is the true ground state electron density.

Kohn and Sham [14] later showed that the problem can be expressed as aset of single particle equations,[

− ℏ2

2m+ Vion(r) + VH(r) + VXC(r)

]ψi(r) = εiψi(r), (2.3)

where the first term on the le hand side again represents the kinetic energy.e second term corresponds to the external potential, the third term is theHartree potential and the last term is the so-called exchange-correlation po-tential.

e Kohn-Sham formulation transforms the fully interacting many-bodyproblem given by (2.1) into a non-interacting system, where each electronmoves within an effective potential. e problem posed by (2.3) is solved nu-merically in a self-consistent way as follows

1. An initial trial electron density n(r) is generated, usually by superposi-tion of atomic densities.

2. Using n(r) the effective potential (expression in the brackets on the lehand side of (2.3)) is evaluated

6

2.2. Exchange-Correlation Functional

3. e Kohn-Sham equation (2.3) is solved to get the single particle wavefunctions ψi(r).

4. e thus obtained wave functions are used to calculate the electron den-sity as n(r) =

∑i ψ

∗i (r)ψi(r)kiki is the occupation

5. e new electron density is compared with the approximated electrondensity. If the two densities are the same then this is the true groundstate electron density. If the two electron densities differ, the density isupdated and the cycle is repeated.

It is still a formidable task to solve the above problem numerically. A host ofalgorithms has been developed to accomplish steps 1-5 as efficiently as possible[10, 12].

2.2 Exange-Correlation Functional

e exact form of the exchange-correlation potential VXC(r), which can beexpressed as a functional derivative of the exchange-correlation energy,

VXC(r) =δEXC(r)

δn(r), (2.4)

is unknown. Over the course of time several exchange-correlation functionals,including for example local density (LDA), and generalized gradient approxi-mations (GGA), have been formulated to study different kind of systems andto achieve maximum chemical accuracy. e simplest approximation is theLDA, which is derived from the homogeneous electron gas. e local densityapproximation for the exchange correlation energy is wrien as

ELDAXC [n(r)] =

∫n(r)ε[n(r)]dr (2.5)

where n(r) is the electron density and ε[n(r)] is the exchange-correlation en-ergy density.

One might ask whether the LDA a good choice for all systems! For bulkmaterial this might be a choice but for atoms, molecules, and interfaces wherethe electron density has large gradients a different exchange-correlation poten-tial is required. LDA are important in the construction of more sophisticatedfunctionals, such as GGA or hybrid functionals. In the study of systems wherethe electron density is not uniform, the exchange-correlation functional can

7


be represented as a GGA, which takes into account the spatial variation of theelectron density in the system

EGGAXC [n(r)] =

∫n(r)ε[n(r),∇n(r)]dr (2.6)

where n(r) is the electron density and ∇n(r) is its gradient.ere is a large family of GGA functionals. In this work all calculations are

performed using the generalized gradient approximation as parameterized byPerdew-Burke-Ernzerhof (PBE) [15].

2.3 Choice of Basis Set

Our objective is to solve the single particle Kohn-Sham equations, (2.3). isrequires to define a basis set, in which the one particle orbitals ϕk(r) are repre-sented. Some possible basis sets include linear combination of atomic orbitals,real space grid, and plane waves. Plane waves are a good choice for periodicsystems and therefore also a good option for our systems. Plane waves havethe advantage that they are numerically very efficient since transformationfrom real to reciprocal space are fast. We can express the single particlel wavefunctions as

ϕk(r) = exp(ik · r)uk(r) (2.7)

where exp(ik · r) is a plane wave function with wave vector k, and uk(r) is aBloch function with the same periodicity as the supercell. e periodicity ofuk(r) means that the single particle wavefunction can be expanded in planewaves as

ϕk(r) =∑G

cG exp(i(k+ G) · r) (2.8)

where the coefficient cG is the probability amplitude corresponding to eachbasis function andG is a reciprocal laice vector. According to this expressionevaluation of G involves a summation over an infinite number of values of Gin the reciprocal space, which has to be cutoff for practical computations. Itis common to set a cutoff for the value of G based on the maximum kineticenergy of any of the plane waves is,

Ecutoff = [ℏ2/2m]G2cut (2.9)

e infinite sum in Eq. (2.8) then becomes

ϕk(r) =Gcut∑G

cG exp[i(k+ G) · r] (2.10)

8

2.3. Choice of Basis Set

We have thus introduced a new parameter, the cutoff energy Ecut, which mustbe defined in a DFT computation. e cutoff energy is element specific socare needs to be taken when dealing with multi element system. Generally thelargest cutoff energy for any of the element in the system is taken as the cutoffenergy for the compound system as demonstrated by the following example.Consider an arbitrary adsorbate on a surface e.g., CTAB on a gold surface. Asupercell of this system contains Au, C, N, Br, and H, for which the plane waveenergy cutoff are given in table below. For this computation the energy cutoffvalue should thus be at least 300 eV.

Table 2.1: Plane wave energy cutoff energies of elements in a compound system.

Element Emax−cut Emin−cut

(eV) (eV)Au 230 172C 400 300N 400 300H 250 200Br 216 162

0.08

0.1

0.12

0.14

0.16

0.18

0.2

0.22

0.24150 200 250 300 350 400 450 500

Sur

face

ene

rgy

(eV

/Å2 )

Plane wave cutoff energy (eV)

(100)(110)(111)(310)

Ecut = 320 eV

Figure 2.1: Convergence with plane wave cutoff energy.

In practice the plane wave cutoff energy (just as any of the other numerical

9


parameters e.g., number of k-points) should be chosen such that the property ofinterest, e.g., laice constant, surface energy or the total energy of the systemare converged to the desired accuracy. Figure 2.1 shows the convergence ofthe surface energy with the plane wave energy cutoff. It demonstrates that acutoff energy of at least 300 eV is required to converge the surface energies. Italso shows that energy differences converge faster than total energies.

2.4 Pseudopotentials

In the description of the basis set, we did not mention valence and core elec-trons and also we did not discuss the description of the wave function in thevicinity of the nucleus. e tightly bound core electrons have wave functionsthat oscillate rapidly close to the nucleus. A large cutoff energy is needed tomake thewave function smooth. A large cutoff energymeans a large number ofplane wave components, which is comptationally very cumbersome. To reducethe burden, pseudopotentials are used, which basically replaces core electronswith a smooth potential. is approximation is justified because from a phys-ical point of view, it is the valence electrons that are important in a reaction.In this thesis work, all DFT computations are done using the projector aug-mented wave method [16, 11], which represents a generalized version of thepseudopotential method and also a connection to the linear augmented planewave method.

2.5 Brillouin Zone Integration

e observables, energy and electron density, of the Kohn-Sham equation (2.3)are calculated by integrating over the first Brillouin zone (BZ). A great deal ofwork reduces to evaluating integrals of the form

Etotal =1

VBZ

∫BZE(k)dk (2.11)

where Etotal is the energy of the ground state and VBZ is the volume of the BZ.e integral in Eq. (2.11) is defined in reciprocal space and integrated over allpossible k values. Numerically this integration is done by replacing the integralwith a summation over a discrete set of k points as described below.

e integral of the function shown in Fig 2.2 is the area under the curve thatis found by integrating in the interval

[−πL, πL

]. A simple way to approximate

this integral is to break up the interval into equal pieces and estimate the areausing the trapezoidal method.

10

2.6. Slab Models of Surfaces

Figure 2.2: Numerically integrating over Brillouin zone

e k-point grid used in DFT calculations is dependent on several factors,such as type of material, size of supercell1, crystal structure, etc. A metal re-quires a different number of k-points than a semiconductor or an insulatorbecause of the difference in the Fermi surface. Another example is the size ofthe supercell as a small supercell in real space corresponds to a large volume inreciprocal space and vice versa. By keeping these things in mind one can savelots of computational time without compromising the result.

2.6 Slab Models of Surfaces

(a) (b)

Figure 2.3: (a) Slab model of (100) surface of Au crystal. (b) Vacuum spacingbetween two neighbouring slabs.

Crystallites are microscopic crystals bound by several facets. Facets innanocrystals are flat surfaces, which appear during crystal growth. us itis vital to start our study with the surface of the crystal. To represent surfacesin our DFT study we analyzed slab models, Fig. 2.3(a). ese models are com-putationally efficient as they are infinite in two dimensions due to application

1cell that contains more than one primitive cell

11


of periodic boundary condition, but finite in the third and yet have all the prop-erties of a real surface. Along the vertical direction slabs are seperated fromeach other by a minimum distance to avoid any kind of interaction betweenthem. e spacing, denoted as vacuum space in Fig 2.3 (b), varies with the typeof adsorbates on the surface. For example, an adsorbate with a long tail wouldrequire a larger vacuum spacing because electrons in the chain would be closerto the next slab.

2.7 Computational Setup

In this thesis work, we have used python and the atomic simulation environ-ment (ASE)[17] to construct the atomistic system. Another tool, which hasbeen used quite extensively in this thesis work, is Ovito [18]. Ovito is a visu-alization tool for atomistic models that proved very handy in the analysis ofstructures.

All DFT calculations are performed using the projector augmented wavemethod [16] as implemented in the Vienna ab-initio simulation package [11]. e exchange-correlation potential used in our DFT calculation is Perdew,Burke, and Ernzerhof (PBE) and is based on the generalized gradient approxi-mation.

In all the calculations involving Au and the adsorbate butyltrimethylam-monium bromide (BTAB), a plane wave cut-off energy of 320 eV was used. Forcomparison also calculations with a cutoff energy of 500 eV were carried out.e difference in the adsorption energy of the system with respect to a cutoffenergy at 320 eV is around 9 meV which is less than kT value at room temper-ature.

Surfaces with several different orientations were considered. e slab mod-els for (100), (110), and (111) contain 10 atomic layers whereas in the case of(310), 6 atomic layers were considered. A vacuum spacing of 12Å was used inall computations.

12

Chapter 3

ermodynamics

3.1 ermodynamics versus Kinetics

e whole of Materials and Surface science can be represented as a bale be-tween thermodynamics and kinetics. is bale is driven by the driving forcesand the energy barriers of the landscape. ermodynamics tells whether a cer-tain transformation is favourable or not, i.e., if there is a decrease in the freeenergy or not. It gives an account of driving forces required for a reaction tooccur. Kinetics is all about, how fast or slow a process can occur i.e., determinesthe rate. e kinetics of a reaction is about how to overcome energy barriersand reach the final state.

e way in which kinetics affect the rate of transformation can be under-stood from Fig. 3.1. e final state is lower in energy than the initial state. Itmeans that the system gains free energy by changing its state from initial to fi-nal. is difference in the free energy∆G is the thermodynamic driving force.However, the barrier, which a particle must cross in order to reach to the finalstate, requires energy from thermal activation. e particle must have energyequal to the activation energy in order to cross it. A particle in an ensemblegains energy by random collisions with other particles and at some point oftime it will gain enough energy to overcome the barrier. is transformationrate can be wrien down as

R = A exp

[− E

kT

], (3.1)

whereE is the activation barrier, k is the Boltzmann constant, T is temperatureand A is a pre-factor. e larger the barrier, the more difficult the process tooccur, the slower the rate. us by varying the heights of the energy barriers,the transformation rate can be controlled. In short, thermodynamics tells us

13

Chapter 3. ermodynamics

Figure 3.1: Transition from initial state (a) to final state (c) through an activationstate (b). Energy barrier is modified when catalyst is used in the reaction.

that a reaction or transformation should take place if the final product is morestable than the initial state. Kinetics concerns the rate of transformation.

A process concerned with kinetics could be a simple chemical reaction, ora transformation from one phase to another. Crystallization, the process bywhich crystals form from solution or from vapour, is a field that is very muchconcerned with kinetics. e shape of crystals is highly influenced by kineticfactors, such as diffusion, adsorption, and desorption. One of the major diffi-culties in understanding the growth mechanism of crystals is the complexityof the energy landscape.

A very generalized picture of a crystal’s physical and energy landscape isshown in Fig. 3.2. An incoming adatom randomly occupies a site on the crys-tal surface. is adatom however, does not remain there for long as it hopsbetween sites and in the process occupies another sites that must be lowerin energy. e particle gets energy for hopping from thermal agitation andfrequent collision with other particles and gradually moves from one state tothe another state. e different energy barriers along the migration path ofthe adatom determine the time scale of transformation. ere are many ex-amples in nature, which are thermodynamically favourable but kinetically un-favourable. Graphite and diamond are both forms of carbon, but graphite hasa lower free energy than diamond. ermodynamically, transformation fromdiamond to graphite is favourable. It is, however, kinetically hindered becausethe transition states are so high in energy that the transition rate is extremelysmall.

14

3.2. ermodynamic Driving Forces

Figure 3.2: Schematic of (a) physical and (b) energy landscape of a surface withone adatom.

3.2 ermodynamic Driving Forces

So what drives a phase transformation? Phase transformation takes the sys-tem to a new state whose free energy is different from the initial state. ischange in free energy provides the necessary driving force. Gibbs free energycomprises contributions from the enthalpy and the entropy of the system. It isdefined as

G = E + pV − TS, (3.2)

where G is Gibbs free energy, E the internal energy of the sytem, p the pres-sure, V the volume, T the temperature, and S the entropy of the system. eE + pV term is the enthalpy H of the system

G = H − TS. (3.3)

Figure 3.3(a) shows a liquid-solid system in equlibrium at the melting tem-perature Tm. e thermodynamic driving force in this case is∆G = 0. It meansthe net transfer of number of particles across the interface is zero. Figure 3.3(b)is the same liquid-solid system but under a finite driving force, which arises forexample if T is slightly lower than Tm. In the second case there exists a drivingforce∆G=GS −GL<0, which favours the transformation. is driving force isdependent on thermodynamic parameters, such as temperature, pressure, com-position, etc. us by varying these parameters it is possible to manipulate thedriving force.

15


Figure 3.3: Liquid-crystal interface (a) at equilibrium and (b) subjected to a finitedriving force.

3.3 Surface Free Energy

e surface energy measures the energy needed to break a bulk material intotwo pieces. In other words the surface energy is the work done when creatinga surface. e surface energy of a single component system is always positive.However, in multicomponent system, due to the interaction between the sur-face and ITS surroundings the surface energy changes and it can even becomenegative [19]. e method used to calculate the surface energy of clean Ausurfaces and Au surfaces with adsorbates is explained below.

e system under study is represented as a slab (see Fig. 2.3) and has sur-faces at the top and boom. e surface energy is obtained by subtracting thebulk energy from the total energy of the slab [20, 21, 22]. In all calculationswe have assumed that the bulk energy of the system is constant for a giventhickness. Mathemetically above statement can be wrien as

γAuhkl =

Eslab − εbulkNslab

Atot

(3.4)

which can be rearranged into

Eslab

Atot

= γAuhkl +

εbulkAtot

Nslab, (3.5)

where γAuhkl is the surface energy of clean Au, Eslab is the total energy of the

slab, εbulkNslab is the bulk energy, and Atot the total surface area.Equation 3.5 corresponds to a straight line whose intercept gives the sur-

face energy of the clean surface and whose slope yields the bulk energy per

16

3.4. ermodynamics of Adsorption

atom. Using this approach we have calculated the surface energies for the lowindexed surfaces (100), (110), and (111) as well as the high indexed surfaces(210), (310), and (410). ese surfaces have been considered because low in-dexed surfaces are generally stable and they appear in almost all crystallinestructures. High indexed surfaces are important because they contain steps,which act as nucleation sites in the process of crystallization.

e free energy of a system containing surface of area A is

G = G0 + Aγ (3.6)

whereG0 is the free energy of the bulk, and γ the surface energy. e increasein the free energy of the system due to an increase in the surface area by dA is

dG = γdA+ Adγ. (3.7)

If we assume that the surface energy is independent of the area of the interfacethen, dγ

dA= 0, which leads to

dG = γdA. (3.8)

3.4 ermodynamics of Adsorption

3.4.1 Adsorbates

Adsorbates are atoms or molecules on a surface, which modify its physical andchemical properties [23, 24]. Apart from modifying the free energy of the sys-tem they also modify many other aspects of the dynamics of the surface. Herewe will consider only the surface energy, which is a purely thermodynamiceffect. e properties of a surface with an adsorbate depend on adsorbate ge-ometry, angular orientation of the adsorbate and the overall structure of sur-face and adsorbate. Adsorbates can adsorb either physically or chemically to asurface:

1. Physisorption: In this mode adsorbates are bonded to the surface viaweak van-der-Waals type forces. eir binding energy is typically com-parable to kT .

2. Chemisorption: Adsorbates in this mode are strongly bonded to the sur-face e.g, by ionic or covalent bonding. e binding energy in this case ismuch higher than kT . is type of bonding is important in the study ofsurfaces because there is substantial redistribution of electrons betweenadsorbate and surface. Chemisorption is almost always exotermic.

17


e rate of adsorption of a molecule onto a surface follows an Arrheniusform

Rads = A exp

[−Ea

kT

](3.9)

where Ea is the activation energy for adsorption, k is the Boltzmann constant,and A is a pre-factor.

(a) (b)

Figure 3.4: (a) (H(CH2)4N(CH3)3+) butyltrimethylammonium and (b)(H(CH2)4N(CH3)3Br) butyltrimethylammonium bromide.

In this work, the adsorption of bromine (Br), butyltrimethylammoniumcation BTA+, and butyltrimethylammonium bromide (BTAB) has been stud-ied on (100), (110), (111), and (310) surfaces. We chose these adsorbates for ourstudy because alkyltrimethylammoniumbromides (CnTAB=CnH2n+1N(CH3)3Br)wheren is the chain length, are themostwidely used surfactant in seed-mediatedgrowth [25, 26, 8] of gold nanoparticles. We have used a shorter chain length(n = 4) in our study to save valuable computational time. e choice ofcounter ion Br− is motivated by the work from Garg and co-workers [7] whoproposed Br− as a suitable candidate because of its moderate binding with thegold surface.

3.4.2 Approximations

e thermodynamic environment, in which Au nanorod growth occurs exper-imentally, is not amenable to direct modeling by DFT due to its complexity.We therefore make a series of approximations as outlined in this section. DFTcalculations are done at T = 0 K whereas real time growth occurs at finite tem-peratures.

Figure 3.5 shows the effect of various thermodynamic parameters on thestructure of a surfactant. e Gibbs free energyG = H−TS of the surfactantcomprises both entropy and internal energy. By bringing the surfactant fromits reservoir, for example from vacuum or solution, onto the surface the change

18

3.4. ermodynamics of Adsorption

Figure 3.5: Schematic visualizing the different thermodynamic contributions thatarise along the transition in the state of surfactant as it is adsorbed from reservoironto the surface at zero K (a) transition in the state of surfactant as it adsorbedfrom the reservoir(vacuum) onto the surface at some finite temperature (b).

in Gibbs free energy is mainly due to the internal energy. At lower temper-atures the entropy is small compared to the internal energy of the surfactant.Solvation effects should also be small compared to the heat of adsorption. Inour calculations the chemical potential of all surfactants is measured with re-spect to the vacuum.

We have used BTAB as a protoype for the surfactant CTAB used experi-mentally. BTAB has only four carbon atoms in its chain compared to 16 in thecase of CTAB. is selection is motivated by the fact that the difference in theadsorption energy for the case of BTAB and CTAB is small [27]. e onlymajorchange is due to weak van-der-Waals interactions between adjacent chains.

3.4.3 Adsorption Sites

e motivation for doing the adsorption study on low index surfaces (100),(110), and (111) is that these surfaces always appear aer nucleation and playa key role in crystal growth. Another advantage is that their DFT computationis quite efficient. High index surfaces, such as (310) are important because oftheir structure. Step atoms on these surfaces have lower coordination, whichmakes these surface more reactive. Evidences [28, 1] have been found for the

19


(a) (100) (b) (110) (c) (111)

hollowon-top

long bridgeshort bridge

hcpon-top

fccbridge

hollowon-top

bridge

Figure 3.6: Adsorption sites on (100), (110), and (111) FCC surfaces.

presence of high indexed surfaces in gold nanorod growth. Indexed surfaceshave lower symmetry and there are many more degrees of freedom for placingadsorbates. Figure 3.6 shows all possible adsorption sites on low index planesof FCC system. It is clear from this figure that there are many possible ad-sorption sites, for example bridge, hollow, on-top, fcc, and hcp. e high indexplane (310) shown in Fig. 4.6 is composed of low index planes. Its terrace has astructure like (100) planes and steps resembles (110) planes.

3.5 Coverage

e surface energy of a clean surface is constant if there is no defect formationor reconstruction of the surface.e situation, however, changes with adsor-bates on the surface. Adsorbates can form strong covalent, metallic, and ionictype bonds with the substrate. Strong bonds change the electronic structureof the substrate close to the surface. Adsorbates change the free energy of thesystem. In the case of a crystal this change is orientation dependent, which isalso called surface energy anisotropy. e surface energy in the presence of anadsorbate is given by

γAu+adshkl = γAu

hkl +∆γadshkl , (3.10)

where ∆γadshkl is the Gibbs free energy of adsorption per unit area, γAuhkl is the

surface energy of the clean surface, and ∆γadshkl is the change in the Gibbs freeenergy per unit area of the system

∆γadshkl =∆Eads − µ

Atot

(3.11)

where ∆Eads is the adsorption energy and µ is the chemical potential of theadsorbate. It is evident from Eq. (3.11) that the chemical potential and thusthe surface energy are dependent on surface coverage. e surface coverage is

20

3.6. Equilibrium Shape

Figure 3.7: Polar plot of surface energy γ(θ) for a square laice. e radius vectorsOA, OB, and OC represents the free energy of a surface plane whose normal liesin that direcrtion. us OA = γ100, OB = γ111. etc. e inner evelope (red line) ofall such Wulf planes gives the equilibrium shape of the crystal.

measured as

adsorbates per surface supercell =number of adsorbate species in supercellnumber of surface atoms in supercell

adsorbates per unit area =number of adsorbate species in supercell

cross sectional area of supercell.

During surfactant assisted crystal growth, surfactants show an affinity to-wards certain crystal planes. A large negative adsorption energy is a signatureof favourable binding to that facet. We saw in the previous section how adsor-bates influence the surface free energy.us it is possible to change the surfacefree energy of different facets at different rates, which represents the basic prin-ciple of anisotropic growth under a finite driving force. We will, however, seethat this is not always the case as there are several other factors which shisthe reaction away from equilibrium.

3.6 Equilibrium Shape

e free energy of a system containing an interface of area A and interfacialfree energy γ per unit area is given by

G = Gbulk + Aγ (3.12)

whereGbulk is the free energy of the bulk and γ is the excess free energy due tobroken bonds close to the surface. e number of broken bonds at the surface

21


Figure 3.8: A truncated cuboctahedral equilibrium shape of an FCC crystalbounded by 111 and 100 facets.

increases as one gradually moves from low to high index planes. A convenientmethod for ploing the variation of surface energy with surface orientation isto construct the polar plot of surface energies γ(θ).

e equilibrium shape of a crystal can be geometrically determined fromthe polar plot of surface energies as suggested by Wulff [29]. e Wulff con-struction, which gives the shape with the minimum

∑γhklAhkl , provides the

eqilibrium shape of a crystal based on variations in surface energy as a func-tion of orientation. e steps involved in the construction can be summarizedas follow

1. Obtain the polar plot of the surface energy from broken bond theory

2. For every point on the γ surface, a plane is drawn through the point andnormal to the radius vector

e equilibrium shape is then simply the inner envelope of all such planes.Crystals under equilibrium obtain a shape which has lowest surface energy

for a given volume. An equilibrium shaped crystal has to satisfy the condi-tion that

∑γhklAhkl be minimum, where γhkl and Ahkl are the interfacial free

energy and the area of hkl planes respectively. Note that the structure ob-tained from the Wulff construction is different from the condition that

∑γhkl

or∑Ahkl be minimum individually.

In this thesis work, we have used Wulffmann soware, a program [29] de-veloped to obtain the equilibrium shape of a crystal. An example for a crystalobtained in this fashon is shown in Fig. 3.8.

22

Chapter 4

Results and Discussion

4.1 Adsorption

4.1.1 Adsorption Sites

In order to study adsorption on any surface it is vital to identify adsorptionsites first. Figure 3.6 shows all possible sites for adsorption on low index FCCplanes. At maximium coverage, a monolayer is formed. Here we assume thatadsorption occurs only on the sites on the surface, not on the top of anotheradsorbate.

4.1.2 Br on Low Index Au Surfaces

In order to find the energetically most favourable site for Br, Br is manuallyincorporated at the sites shown in Fig. 3.6 and their adsorption energies arecompared. e results are summarized in Table 4.1 which shows that the bridgesite is energetically most favourable site. e same approach is taken to findthe most stable sites for Br on (110) and (111) surfaces. For the case of (110)surface, bridge site is found most stable and for (111) surface, it is the fcc site.

To illustrate the effect of the surface coverage let us consider the adsorptionof Br at hollow sites on (100) Au surfaces. e different coverages are repre-sented by size of the supercell. A large supercell, for example 4×4 surface unit

Table 4.1: Adsorption energies of Br for various sites on (100) Au surface at fullcoverage.

Adsorption site on-top hollow bridge

Adsorption energy Eads (eV) −0.75 −0.99 −1.08

23

Chapter 4. Results and Discussion

(a) (b) (c)

Figure 4.1: Br on various FCC Au surfaces. 1 × 1 supercells of (a) (100) (b) (110),and (c) (111) surfaces.

(a) (b) (c) (d)

Figure 4.2: Br on FCC Au (100) hollow sites at different coverages. (a) 1 × 1supercell (1 monolayer = 1 adsorbate per substrate atom), (b) 2×2 (1/4 monolayer),(c) 3× 2 (1/6 monolayer), and (d) 4× 4 (1/16 monolayer coverage).

Table 4.2: Adsorption energy of Br on Au (100) surfaces with respect to surfacecoverage. e surface coverage is expressed in terms of monolayer (ML).

Coverage (ML) 1 1/2 1/4 1/6

Adsorption energy Eads (eV) −0.02 −0.46 −1.05 −1.09

24

4.1. Adsorption

Table 4.3: Adsorption energies of BTA+ at various sites on Au (100) surfaces atfull coverage.

Adsorption site on (100) EBTA+

ads

on-top −1.90hollow −1.86bridge −1.88

(a) (b)

Figure 4.3: (a) BTA+ cation on the Au (100) surface on-top site. (b) Triangularshape of the base of the BTA+ cation. Nitrogen is represented in blue, carbon ingrey, and hydrogen in white.

cells, represents very low coverage. From Table 4.2 we can see that the adsorp-tion energy is becoming more negative as we go from high surface coverage tolow coverage. Comparing the adsorption energies at coverages of 1/4 ML and1/6 ML we see that the difference in the adsorption energy is only −0.04 eV.is is an indication that the dilute limit has been reached and there will notbe any further change in the adsorption energy at lower coverages.

4.1.3 BTA+ on Low Index Au Surfaces

In the following the position of the BTA+ cation is measured with respect tothe position of the nitrogen relative to the surface site. e adsorption energiesare summarized in Table 4.3. From this table it is evident that the on-top siteon the (100) surface is the most favourable site for BTA+. e same approachis taken to find the most stable sites on (110) and (111) surfaces. e on-top siteis found to be the most stable site in all cases. is probably due to the electrondeficient cation BTA+ favouring to sit close to an Au atom.

25


Table 4.4: Adsorption energy of BTA+ at full coverage incorporated at the moststable site.

Surface Adsorption site Adsorption energy (eV)

(100) on-top −1.76(110) on-top −3.36(111) on-top −1.43

Figure 4.4: BTAB surfactant on FCC gold surface (100). BTA+ is located at theon-top site in a 2× 2 supercell and Br at all possible sites in the supercell.

4.1.4 BTAB on Low Index Surfaces

Now that we have found the most stable sites for Br and BTA+ on low indexsurfaces, we continuewith the analysis of BTAB on low index surfaces. BTA+ isfirst placed on the on-top site which is the most favourable site for BTA+ basedon the result from the previous subsection. Br is then placed at all available sitesin the supercell see Fig. 4.5. Utilizing the symmetrical structure of BTA+ a fewsites are excluded from the computation. From Table 4.5 it is evident that themost stable structure of BTAB on Au (100) is with BTA+ at the on-top site andBr at the bridge site.

26

4.1. Adsorption

Table 4.5: Relative energies of a 2×2 supercell of Au (100) surface with one BTABmolecule. e numbers in the first column are the corresponding sites for Br inthe supercell as shown in Fig. 4.4.

Site 1 2 3 4 5 6 7 8 9

Energy (eV) −6.95 −6.90 −6.92 −0.01 −3.01 −6.92 −7.00 0.00 −7.10

Figure 4.5: Adsorption sites on (310) Au surfaces. e supercell contains oneprimitive unit cell shown by the black rectangle. Atoms in blue are step atoms,light yellow atoms are terrace atoms and dark yellow are bulk atoms. All crystal-lographically distinct possible adsorption sites are marked with numbers.

Figure 4.6: Side view of Au (310) surface with step atoms highlighted in blue.

27


Table 4.6: Relative energies of a 1 × 1 supercell of Au (310) surface with one Br.e numbers in the first column are the corresponding sites for Br in the supercellas shown in Fig. 4.5.

Site 1 2 3 4 5 6 7 8

Energy (eV) −0.15 −0.16 −0.09 −0.16 −0.07 −0.08 0.00 −0.15

Table 4.7: Relative energy of a 1×2 supercell of Au (310) surface with oneBTA+

cation. e numbers in the first column are the corresponding sites for BTA+ inthe supercell as shown in Fig. 4.5.

Site 1 2 3 4 5 6 7 8

Energy (eV) −0.19 −0.15 0.00 −0.16 −0.20 −0.24 −0.10 −0.03

4.1.5 Br on High Index Surface

Along with the low index surfaces, the high index (310) surface is also con-sidered in this study. High index surfaces are chosen because of the presenceof surface steps, which provide reactive sites and are seen as nuceation sitesduring crystal growth. Br is placed at all the sites shown in Fig. 4.5 and thetotal free energy of the system is calculated. e results shown in Table 4.6demonstrate that the step sites, indicated by numbers 4 and 9 in Fig. 4.5, areenergetically the most favourable sites for Br.

4.1.6 BTA+ on High Index Surface

We use Fig. 4.5 to identify the adsorption sites for BTA+ on the (310) surface. Asupercell of size 1× 2 is used in the analysis because of the large size of BTA+.e results shown in Table 4.7 reveal the most favourable site for BTA+ on the(310) surface. e most favourable site for BTA+ is site number 6 as shown in

Table 4.8: Two possible orientations of BTA+ on (310) with respect to the step,which is shown in dark yellow. e adsorption energy is calculated for each ge-ometry and tabulated in the second column.

Orientation

Adsorption energy (eV) −2.71 −2.82

28

4.2. Surface Energy

Table 4.9: Four possible orientations of BTAB on (310) Au surface with respectto the step site, which is shown in dark yellow. ese four geometries have beenconsidered based on the orientation of the tail with respect to the step and theposition of Br with respect to the tail.

Orientation

Adsorption energy (eV) −1.11 −1.23 −1.45 −1.43dN−Br (Å) 4.98 7.64 4.72 4.94

Fig. 4.5, which is between the two step sites represented by numbers 4 and 9.In next step, we take into account the rotation of BTA+ with respect to its tail.In one orientation, the tail of BTA+ is pointing towards the step while it facesin the opposite direction in the second.

e difference in the energy for the above case is approximately 0.1 eV,which is higher than the thermal energy (0.026 eV) at room temperature. isdifference in the adsorption energy is due to the position of chain atoms withrespect to the step site. e chain atoms in one orientaion are closer to thestep than the other. It is the interaction between the atoms in the tail of thesurfactant and the step atoms, which makes one structure stable than the other.

4.1.7 BTAB on High Index Surface

From the previous subsections on Br and BTA+ on high index surfaces, we nowknow the most stable sites for Br and BTA+ on (310) surface. e most stablesite for Br is near the step and for BTA+ it is on the terrace. Now we find themost stable orientation of BTA+ with respect to the step site in the presence ofBr. e results are summarized in Table 4.9 and reveal themost stable geometryof BTAB on (310) surface. In this conffiguration BTA+ and Br sit close to eachother.

29


−70

−60

−50

−40

−30

−20

−10

0

10

0 5 10 15 20

Sla

b e

ne

rgy (

eV

)

Number of atomic layers (N)

(y=ax+b)

best fit result: a= −3.2230 b = 0.9620

fitDFT data

b = surface energy of (100)

Figure 4.7: Surface energy of FCC gold (100). e abscissa yields the (100) surfaceenergy.

Table 4.10: Surface energy of high and low index FCC Au surfaces.

hkl (surface) γhkl(J/m2)

(100) 0.8881(110) 0.9984(111) 0.7333(210) 1.0215(310) 0.9396(410) 1.0106

30

4.2. Surface Energy

4.2 Surface Energy

4.2.1 Surface Energies of Clean Au Surfaces

e surface energy of an N atomic layer thick clean gold surface can be ex-pressed as described in Sect. 3.3 as

Eslab

Atotal

= γAuhkl +

εbulkAtot

Nslab,

where Eslab is the total energy of the slab and Atotal is the combined area oftop and boom surfaces. Equation (3.5) corresponds to a straight line whoseintercept is the surface energy of the gold surface and the slope is the bulkenergy per atom. e surface energy is obtained by fiing as shown in Fig. 4.7.

Using this approach, the surface energies of several gold surfaces havebeen calculated as tabulated in Table 4.10. It is evident from these data that(111) is the most stable surface whereas for example (110) is energetically lessfavourable. is result can be explained by the broken bond model. In creatinga FCC (110) surface, five bonds per unit cell are broken, whereas there are fourin the case of (100) and only three in the case of (111). e (111) surface is aclosely packed smooth surface, whereas (110) is an open surface with manyundercoordinated atoms.

4.2.2 Surface Energies Change due to Adsorption

In this section we study the change in Gibbs free energy of the system dueto adsorption of Br, BTA+, and BTAB. e total surface energy in the case ofadsorption is

γAu+adshkl = γAu

hkl +∆γhkl (4.1)

where ∆γhkl is the Gibbs free energy change per unit area, γAuhkl the surface

energy of the clean surfaces, and γAu+adshkl is the surface free energy post ad-

sorption.

Br Adsorption

Figure 4.8 shows the change in the Gibbs free energies of various gold surfacesin the presence of Br. e change in the energy density depends on the chem-ical potential of Br, which represents the potential of the reservoir. Here wereference µ with respect to the Br dimer in vacuum, which sets the zero of theenergy scale, µ = Edimer +∆µ. It is clear from Fig. 4.8 that the change in thefree energy according to Eq. (4.1) is different for different surfaces.

31


−120

−100

−80

−60

−40

−20

0

20

−3 −2 −1 0 1 2 3

Gib

bs fr

ee e

nerg

y ch

ange

(m

eV/Å

2 )

Chemical potential wrt dimer in vacuum (eV)

(3×2)(2×2)(2×1)(1×1)

Br richBr poor

(a) Br on FCC gold (100) surface

−120

−100

−80

−60

−40

−20

0

20

−3 −2 −1 0 1 2 3

Gib

bs fr

ee e

nerg

y ch

ange

(m

eV/Å

2 )


(3×2)(2×2)(1×1)

Br richBr poor

(b) Br on FCC gold (110) surface

−120

−100

−80

−60

−40

−20

0

20

−3 −2 −1 0 1 2 3

Gib

bs fr

ee e

nerg

y ch

ange

(m

eV/Å

2 )


(3×2)(2×2)(2×1)(1×1)

Br richBr poor

(c) Br on FCC gold (111) surface

−120

−100

−80

−60

−40

−20

0

20

−3 −2 −1 0 1 2 3

Gib

bs fr

ee e

nerg

y ch

ange

(m

eV/Å

2 )


(2×2)(1×2)(1×1)

Br richBr poor

(d) Br on FCC gold (310) surface

Figure 4.8: Gibbs free energy change of Au surfaces with Br as adsorbate at dif-ferent surface coverages. e chemical potential of Br is measured with respectto the dimer in vacuum. Different line styles correspond to different surface cellsand thus different coverages, for example 1× 1 represents the highest coverage.

32

4.2. Surface Energy

0

10

20

30

40

50

60

70

−3 −2 −1 0 1 2 3

Sur

face

free

ene

rgy

(meV

/Å2 )


Br radical

(100)(110)(111)(310)

Br richBr poor

Figure 4.9: Variation in the surface free energies of various FCC gold surfaces inthe presence of Br. e chemical potential of Br is measured with respect to thedimer in vacuum. Different line styles represent different surfaces.

By combining the data from Fig. 4.9 with the values for the energy densityof clean Au surfaces from Table 4.11 we obtain the total surface free energyfrom Eq. (4.2). Figure 4.10 gives information about the relative stability ofsurfaces. On the extreme le of the plot in Fig. 4.10, which represents cleansurfaces shown as flat lines, we can see that the (111) surface is by far themost stable. However, with the increase in the chemical potential, the (110)surface transforms into a more stable state and it even equals the most stablesurface (111) close to ∆µ = 0. Note that the data in Fig. 4.10 implies that thecrystal becomes unstable (γ < 0) with respect to surface formation already at∆µ < 1 eV below the Br radical.

BTA+ Adsorption

Figure 4.11 shows the change in the Gibbs free energies of various gold surfacesdue to the presence of BTA+. e change in the energy density has the samedependence on the chemical potential of BTA+ as explained in the previoussubsection for the case of Br. e main difference in the adsorption of BTA+

and Br on these surfaces is the large change in the free energy of (110) withBTA+. is is due to the large adsorption energy of BTA+ on the (110) surfaceas shown in Table 4.4.

e total surface free energies with respect to the chemical potential ofBTA+ have been calculated the same way as explained for the case of Br. Wecan see in Fig. 4.11 that the surface energy of (110) is decreasing much fasterthan for the other surfaces. is is because of the large change in the Gibbs free

33


−80

−70

−60

−50

−40

−30

−20

−10

0

10

−3 −2 −1 0 1 2 3

Gib

bs fr

ee e

nerg

y ch

ange

(m

eV/Å

2 )

Chemical potential wrt BTA+ in vacuum (eV)

(4×3)(3×2)(2×2)

BTA richBTA poor

(a) BTA+ on gold (100) surface

−80

−70

−60

−50

−40

−30

−20

−10

0

10

−3 −2 −1 0 1 2 3

Gib

bs fr

ee e

nerg

y ch

ange

(m

eV/Å

2 )


(3×2)(2×2)

BTA richBTA poor

(b) BTA+ on gold (110) surface

−80

−70

−60

−50

−40

−30

−20

−10

0

10

−3 −2 −1 0 1 2 3

Gib

bs fr

ee e

nerg

y ch

ange

(m

eV/Å

2 )


(4×3)(3×2)(2×2)

BTA richBTA poor

(c) BTA+ on gold (111) surface

−80

−70

−60

−50

−40

−30

−20

−10

0

10

−3 −2 −1 0 1 2 3

Gib

bs fr

ee e

nerg

y ch

ange

(m

eV/Å

2 )


(2×2)(1×2)

BTA richBTA poor

(d) BTA+ on gold (310) surface

Figure 4.10: Gibbs free energy change of Au surfaces with BTA+ as adsorbateat different surface coverages. e chemical potential of BTA+ is measured withrespect to BTA+ in vacuum. Different line styles represent different surface cells,for example 1× 2 represents the highest coverage.

34

4.3. Equilibrium Crystal Shape

0

10

20

30

40

50

60

70

−3 −2 −1 0 1 2 3

Sur

face

free

ene

rgy

(meV

/Å2 )

Chemical potential wrt BTA+ in vacuum(eV)

(100)(110)(111)(310)

BTA richBTA poor

Figure 4.11: Variation of the surface free energies of various FCC gold surfaces inthe presence of BTA+. e chemical potential is measured with respect to BTA+

in vacuum. Different line styles represent different surfaces.

energy of (110) surface aer adsorption of BTA+. is is an important resultbecause (110) lowers its surface energy at a much faster rate and becomes asstable as (111) at lower chemical potential compared to for example Br.

BTAB Adsorption

Figure 4.14 shows the transition in the stability of various FCC gold surfaces.Surfaces, which are not so energetically favourable in the clean state, for ex-ample (110), become increasingly stable when the chemical environment ischanged. e effect of BTAB on the high energy planes is evident in Fig. 4.14,where one can see that (110) and (111) have equal surface energy at a chemicalpotential −1 eV. Note that the (110) surface transforms into a thermodynami-cally stable plane under the influence of BTAB.

4.3 Equilibrium Crystal Shape

We have used theWulff construction technique[29] to determine the minimumenergy crystal shape see section (3.6). e input quantities are the point groupsymmetry of the crystal and the surface energies of different facets. e spacegroup symmetry used to determine the shape of cubic crystals is Fm3̄m andthe surface energies are taken from section (4.2).

Figure 4.15 shows the variation of the total surface free energies with thechemical potential of Br and shapes of the crystal determined at different chem-ical potentials. It is evident from Fig. 4.15 that the morphology of the crystal is

35


−40

−30

−20

−10

0

10

−3 −2 −1 0 1 2 3

Gib

bs fr

ee e

nerg

y ch

ange

(m

eV/Å

2 )

Chemical potential wrt BTAB in vacuum (eV)

(4×3)(3×2)

BTAB richBTAB poor

(a) BTAB on FCC gold (100) surface

−40

−30

−20

−10

0

10

−3 −2 −1 0 1 2 3

Gib

bs fr

ee e

nerg

y ch

ange

(m

eV/Å

2 )


(4×3)(3×2)

BTAB richBTAB poor

(b) BTAB on FCC gold (110) surface

−40

−30

−20

−10

0

10

−3 −2 −1 0 1 2 3

Gib

bs fr

ee e

nerg

y ch

ange

(m

eV/Å

2 )


(4×3)(3×2)

BTAB richBTAB poor

(c) BTAB on FCC gold (111) surface

−40

−30

−20

−10

0

10

−3 −2 −1 0 1 2 3

Gib

bs fr

ee e

nerg

y ch

ange

(m

eV/Å

2 )


(2×2)(1×2)

BTAB richBTAB poor

(d) BTAB on FCC gold (310) surface

Figure 4.12: Gibbs free energy change of Au surfaces with BTAB as adsorbateat different surface coverages. e chemical potential of BTAB is measured withrespect to BTAB in vacuum. Different line styles represent different cells, for ex-ample 1× 2 represents the highest coverage.

36


0

10

20

30

40

50

60

70

−3 −2 −1 0 1 2 3

Sur

face

free

ene

rgy

(meV

/Å2 )


(100)(110)(111)(310)

BTAB richBTAB poor

Figure 4.13: Variation of the surface free energies of various FCC gold surfaces inthe presence of BTAB. e chemical potential is measured with respect to BTABin vacuum. Different line styles represent different surfaces.

changing with the variation in the surface energy of the surfaces. One can seethis effect by looking at the morphology at chemical potentials 0 eV and 0.8eV. Surfaces which appear in the Wulff shape are thermodynamically stable.Surface (110) is not seen in the morphology, which is because this surface hasa significantly higher surface energy than either (111) or (100).

Figure 4.16 shows the variation of the surface free energies of FCC Au withrespect to the chemical potential of BTAB. It also shows themorphologies of thecrystal at different surface energies. We can clearly see the morphology chang-ing with the surface energies of the surfaces. Facets which are not present inthe equilibrium shape, for example (110) appears later when the surface freeenergy is lowered. For example one can clearly see how (110) grows in size asthe chemical potential is changed from -1 eV to −0.8 eV.

ese results clarify the effect of the chemical environment on the mor-phology of the crystal. Certain facets, e.g. 110 appear and grow in size as theyare stabilized in the chemical environment. ermodynamic stability of a par-ticular plane is determined by whether that plane appears on the Wulff shapeor not.

37


0

10

20

30

40

50

60

70

�3 �2 �1 0 1 2 3

Sur

face

free

ene

rgy

(meV

/Å2 )


Br radical

(100)(110)(111)(310)

Br richBr poor

Equilibrium crystal shape

Figure 4.14: Wulff shape of FCC Au crystal with Br as surfactant. Crystal isbounded by surfaces, (111) in yellow, (100) in blue. Equilibrium shape is also dis-played which is calculated from the surface energies of clean Au surfaces.

38


0

10

20

30

40

50

60

70

80

3 2 1 0 1 2 3

Sur

face

free

ene

rgy

(meV

/Å2 )

(100)(110)(111)(310)

BTAB richBTAB poor

crystal morphologygreen- {110}yellow-{111}blue- {100}

Equilibrium crystal shape

Figure 4.15: Wulff shape of FCC Au crystal with BTAB as the stabilizer. Crystalis bounded by surfaces, (111) in yellow, (100) in blue, and (110) in green. eEquilibrium shape calculated from the surface energies of clean Au surfaces isshown on the far le.

39


40

Chapter 5

Discussion and Conclusion

5.1 Template Assisted Growth

Electrochemical deposition in templates and solution based seedmediated growthare two key methods for the synthesis of anisotropic crystals. Solution-basedmethods requires precise tuning of nucleation and growth steps to achieve thedesired morphology due to complexity of the growth environment . Nanopar-ticle growth is basically a two step process including nucleation and subse-quent growth of nuclei into crystal. Crystal nucleation in solution is largelycontrolled by thermodynamics. eir shape is dictated by the surface energyanisotropy which is dominant on that scale. Seed crystals aggregate and thenbreak into individual particle multiple times until they reach a critical sizewhich is thermodynamically stable and from where it continues to grow.

e growth mechanism of gold nanorods, a template assisted growth assuggested by Murphy et al. [2, 6], where a rod like template (bilayer surfactantstructure) is considered as the main driving force for the nanorod formation,which is a case of heterogeneous nucleation. is mechanism, however, isquestioned by Garg et al. [7] as they showed that nanorod growth is possibleeven without any template formation in the solution. ey also argued that ifthe miceller template is indeed the reason for nanorod formation, one shouldnot obtain Au nanorods at high yield when CTAB concentration is below itscmc value. us the aforementioned arguments clearly indicate that nanorodsare not entirely a product of just a template.

5.1.1 Chemical Potential

e thermodynamic environment, in which Au nanorod growth occurs, can-not be reproduced by DFT due to its complexity. It is, however, possible tostudy the nanorod growth by making a series of well approximations. ese

41

Chapter 5. Discussion and Conclusion

approximations are motivated by the consideration of the magnitude of dif-ferent contributions as shown in Fig. 5.1. e set of approximations, whichwe have made in this study are as follow. Entropic contributions and solva-tion effects have been neglected. We have used BTAB as a prototype for thesurfactant CTAB by ignoring weak van-der-Waals interaction.

Figure 5.1: Schematic visualizing the different thermodynamic contributions thatarise along the transition in the state of surfactant as it is adsorbed from reservoironto the surface at zero temperature (a) transition in the state of surfactant as itadsorbed from the reservoir (vacuum) onto the surface at some finite temperature(b).

e schematic Fig. 5.1 shows the major thermodynamic parameters. eGibbs free energy G = H − TS of the system comprises both the entropy andinternal energy. us by bringing the surfactant from its reservoir (vacuum,solution, etc.) onto the surface, the change in Gibbs free energy is mainly dueto its internal energy. At lower temperature the associated entropy term issmall compared to the internal energy of the surfactant. Solvation effects aresmall compared to the heat of adsorption.

5.1.2 Anisotropic Shapes and Symmetry Breaking

Anisotropic crystal growth (see Fig. 5.2) cannot be explained by thermody-namics alone. e rate at which monomer crystal units aach to facets andmove on these surfaces dictates the non-equilibrium morphology. e rate at

42

5.1. Template Assisted Growth

Figure 5.2: Growth of gold crystal under finite driving force. Equilibrium shape(a) is denoted as cuboctahedral while the final shape (b) is in the form of a rod.

which these units migrate on growing facets in turn depends on the energylandscape of the surface. Under a finite driving force two limiting cases can bedistinguished as the morphology is controlled by either the adsorption or thesurface diffusion rate. e kinetic control hypothesis has been one of the mostsuccessful in describing the non-equilibriummorphology of crystals. emaincriterion to achieve kinetic control is that the reaction should proceed awayfrom equilibrium. e deviation from eqilibrium is rationalized by the changein free energy of crystal facets during the reaction. Figure 5.2 is a good ex-ample of kinetically controlled crystal growth where the addition of monomerunits to the facets is much slower because of several activation barriers in itsmigration path. Under such conditions, the end product assumes a shape thatdeviates from the thermodynamically favoured equilibrium shape.

Under kinetically controlled growth of crystals it is possible to get a varietyof crystal morphologies. is can be accomplished by causing the differencein surface energies of different planes by selective adsorption of surfactants ondifferent facets. ere are, however, few bewildering morphologies of crystalswhich cannot be explained entirely based on thermodynamics or kinetics [30].

e growth of an equilibrium shaped cuboctahedral crystal into a nanorodis an example of a symmetry breaking phenomenon. In the case of symmetrybreaking growth is favoured along one out of many symmetry equivalent di-rections. is class of growth is perhaps instigated by surfactants, for exampleby oriented aachment of surfactants on crystal facet or by using surfactants asa so template for the directed growth. Surfactants could amplify the growthrate on a certain facet, those facet grow much faster compare to other facet

43


Figure 5.3: Representation of growth morphologies in which symmetry is notbroken (le) and with broken symmetry (right). e equilibrium shape of crystalis cuboctahedral while the final shape is in the form of a rod when the symmetryis broken and prism shaped when symmetry is conserved.

which leads to the formation of one dimensional structures as shown in Fig.5.3.

5.2 Conclusion

In this thesis work we studied the thermodynamics of growth of gold nanopar-ticles. We considered (100), (110), (111), and (310) facets, which was motivatedby high-resolution transmission electron microscope [31] images indicatingthat the nanoparticles are bound by the (100), (110), and(111) facets. e equi-librium shape of gold crystals obtained by combining density functional theorycalculations and the Wulff construction method was found as cuboctahedral inagreement with the literature [30]. is result is a demonstration that nucle-ation of any crystal is thermodynamically guided by the predominance of thesurface energy term, which dictates the stability and energy of the nucleus.However, growth of the crystal cannot be described by thermodynamics alone.

We have also shown that adsorption of surfactants, specifically Br, andBTAB leads to a considerable reduction of the surface energies. Density func-tional theory calculations of the interaction of BTAB on gold nanoparticle withseveral facets shows that the binding of BTAB with (110) is energetically muchfavourable compared to other low index planes (100), and (111). is leads toa marked reduction of the surface energy of (110) surface with respect to (111)which is by far the most stable surface in the absence of surfactants. is mightlead to oriented aachment of BTAB adsorbates on (110) and inhibit growth ofthis facet.

e Gibbs free energy change of an interface upon adsorption is dependent

44

5.2. Conclusion

on the chemical environment. It depends on the chemical potential of the reser-voir. We have been able to show that by changing the chemical potential, itis possible to modulate relative surface energies of interfaces and thus a widerange of morphology is possible. Furthermore, the thermodynamic stabilityof specific crystal plane determines whether that plane appears on the Wulffshape or not.

45


46

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Thermodynamics of gold nanoparticle growth